<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> implies  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a></span><br/>

<b>let </b><font color="Maroon" title="c2">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:29"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E4:29"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:29"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="seq_2.html#T23" title="SEQ_2:th.23">SEQ_2:23</a></i>;<br/>
<a NAME="E5:29"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:29_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="pscomp_1.html#E3:29_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E6:29_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E4:29_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></i>;<br/>
<a NAME="E7:29_1"/>
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E8:29_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:29"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E6:29_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:29"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E4:29"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a></i>;<br/>
<a NAME="E7:29"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:29"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="seq_2.html#T24" title="SEQ_2:th.24">SEQ_2:24</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E6:29"><i><font color="Green" title="E24">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:29"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
